LORENTZ FACTOR DERIVATION

The math of the Michelson-Morley experiment follows. For the details
leading up to the math see
The Michelson-Morley Experiment.
Note that Michelson uses "V" rather than "c" for the velocity of light.
Apparently, this was the convention in 1887.

Let:
V = velocity of light.
v = velocity of the earth in its orbit.
D = distance ab or ac, Fig. 72.
T = time light occupies to pass from a to c.
T1 = time light occupies to return from c to
a1 (Fig. 73).

Then T = D / (V - v) and T1 = D / (V + v)

The whole time going and coming is
T + T1 = 2D [V / (V2 - v2)],

and the distance traveled in this time is

2D [V2 / (V2 - v2)] =
2D [1 + (v2 / V2)]

{Note that in a true equation, the right side becomes
2D [1 /(1 - v2 / V2)]
But Michelson and Morley knew that this was approximately
the same as 2D [1 + (v2 / V2)]
within the limits of their measurements.}

Neglecting the terms of the fourth order, the length of the other
path is evidently

2D [1 + (v2 / V2)]1/2,

or to the same degree of accuracy,
2D [1 + (v2 / 2V2)].

The difference is therefore D (v2 / V2).

If now the whole apparatus be turned through 900, the
difference will be in the opposite direction, hence the displacement
of the interference fringes should be

2D (v2 / V2).

Fitzgerald and Lorentz

Fitzgerald and Lorentz independently suggested that motion might
cause a change of length in the direction of motion. The precise
change can be shown very simply by making one distance equal to
the other multiplied by a factor shown here as "X":

If we substitute the modern "c" for "V", and invert the result
(which would have been the case had we originally placed "X" on the
right side of the equation),
we have the Lorentz
factor as it is commonly seen today.

1 / [1 - (v2 / c2)]1/2

Bear in mind that this was not the reason for the low results of the
Michelson-Morley experiment or of low results of the ones of a
similar nature
which followed. The details can be found be reading Is There
a Dynamic Ether on this website.

Note that the fumbling around to arrive at the Lorentz factor back then
was considerably more complicated than postulating a dynamic ether
as was done in nether theory. See Time Dilation and
Ether Relativity on this website.